Exponential Separation of Quantum and Classical One-Way Numbers-on-Forehead Communication

TL;DR

This paper proves the first exponential separation between quantum and classical one-way NOF communication complexity by analyzing a lifted Hidden Matching problem, demonstrating quantum efficiency versus classical limitations.

Contribution

It establishes the first explicit exponential separation in one-way NOF communication complexity between quantum and classical protocols.

Findings

Quantum protocol for the problem has $O(\log n)$ cost.

Classical randomized protocols require $\Omega(n^{1/3}/2^{k/3})$ communication.

Separation applies even with multiple communicating players.

Abstract

Numbers-on-Forehead (NOF) communication model is a central model in communication complexity. As a restricted variant, one-way NOF model is of particular interest. Establishing strong one-way NOF lower bounds would imply circuit lower bounds, resolve well-known problems in additive combinatorics, and yield wide-ranging applications in areas such as cryptography and distributed computing. However, proving strong lower bounds in one-way NOF communication remains highly challenging; many fundamental questions in one-way NOF communication remain wide open. One of the fundamental questions, proposed by Gavinsky and Pudl\'ak (CCC 2008), is to establish an explicit exponential separation between quantum and classical one-way NOF communication. In this paper, we resolve this open problem by establishing the first exponential separation between quantum and randomized communication complexity in one-way NOF model. Specifically, we define a lifted variant of the Hidden Matching problem of Bar-Yossef, Jayram, and Kerenidis (STOC 2004) and show that it admits an ($O(\log n)$)-cost quantum protocol in the one-way NOF setting. By contrast, we prove that any $k$-party one-way randomized protocol for this problem requires communication $\Omega(\frac{n^{1/3}}{2^{k/3}})$. Notably, our separation applies even to a generalization of $k$-player one-way communication, where the first player speaks once, and all other $k-1$ players can communicate freely.